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"Francis William Lawvere was one of the most influential figures in the late 20th\ncentury and up till now, because of\nhis drive to unify and simplify mathematics, by sharpening the\ntools of category theory. The following is an attempt of describing\nsome of the milestones and visions in this process."
]
}
],
"identifiers": [],
"references": [
{
"type": "Article",
"id": "bib-bib1",
"authors": [],
"title": "\nM. Artin, A. Grothendieck and J. L. Verdier,\nThéorie des topos et cohomologie etale des schémas. Tome 1: Théorie des topos.\nLecture Notes in Math. 269, Springer, Berlin (1972)\n"
},
{
"type": "Article",
"id": "bib-bib2",
"authors": [],
"title": "\nM. M. Clemetino and J. Picado,\nInteview with F. William Lawvere.\nhttp://www.mat.uc.pt/~picado/lawvere/interview.pdf\n(2007)\n",
"url": "http://www.mat.uc.pt/~picado/lawvere/interview.pdf"
},
{
"type": "Article",
"id": "bib-bib3",
"authors": [],
"title": "\nB. Eckmann (ed.), Seminar on triples and categorical homology theory (ETH 1966/67). Lecture Notes in Math. 80, Springer, Berlin (1969)\n"
},
{
"type": "Article",
"id": "bib-bib4",
"authors": [],
"title": "\nM. P. Fourman, C. J. Mulvey and D. S. Scott (eds),\nApplications of sheaves. Proceedings of the research symposium on applications of sheaf theory to logic, algebra and analysis (Durham 1977), Lecture Notes in Math. 753, Springer, Berlin (1979)\n"
},
{
"type": "Article",
"id": "bib-bib5",
"authors": [],
"title": "\nA. Kock (ed.),\nTopos theoretic methods in geometry,\nVarious Publications Series 30, Aarhus University, Aarhus (1979)\n"
},
{
"type": "Article",
"id": "bib-bib6",
"authors": [],
"title": "\nF. W. Lawvere (ed.),\nToposes, algebraic geometry and logic.\nLecture Notes in Math. 274, Springer, Berlin (1972)\n"
},
{
"type": "Article",
"id": "bib-bib7",
"authors": [],
"title": "\nF. W. Lawvere, Axiomatic cohesion.\nTheory Appl. Categ. 19, no. 3, 41–49 (2007)\n"
},
{
"type": "Article",
"id": "bib-bib8",
"authors": [],
"title": "\nF. W. Lawvere and S. H. Schanuel (eds.),\nCategories in continuum physics.\nLecture Notes in Math. 1174, Springer, Berlin (1986)\n"
},
{
"type": "Article",
"id": "bib-bib9",
"authors": [],
"title": "\nF. W. Lawvere and S. H. Schanuel,\nConceptual mathematics.\nCambridge University Press, Cambridge (1997) (2nd ed. 2009)\n"
}
],
"title": "F. William Lawvere (1937–2023): A lifelong struggle for the unity of mathematics",
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"1 Continuum physics"
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"content": [
"Lawvere was born in February 1937, as son of a farmer in Muncie, Indiana. He studied\nphysics at the University of Indiana, and there soon felt the need for more useable\nand explicit foundations for the reasoning employed, in particular in continuum physics.\nHe was in Indiana a student of Clifford Truesdell, the founder of the Springer journal\n“Archive for Rational Mechanics and Analysis,” who had a similar foundational agenda.\nL. saw already at this time the need for a category-theoretic approach. One first step\nwas to achieve a “categorical dynamics” (some of which was materialized in the late 1960s).\nA crucial step was his category-theoretic formulation of the formation of function spaces,\nin terms of universal properties (adjoint functors): Cartesian closed categories."
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"F. William Lawvere, Braga, March 2007 (© M. M. Clementino and J. Picado)"
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"Truesdell personally contacted Eilenberg to facilitate L.’s entrance into Columbia as Eilenberg’s Ph.D. student 1960–63 – with a break 1961–63, where L. went to California,\nto learn more set theory and logic from experts in the fields (Tarski, Scott and others). In the California period, L. finished his (Columbia) Ph.D. thesis on functorial semantics of algebraic theories, where in particular\nthe notion of algebraic theory was given in a presentation-free way."
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"2 The Category of Sets"
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"For L. himself, a turning point in his general search for useable and teachable foundations\nfor mathematics was the year 1963–64 as an assistant professor at Reed College in Oregon.\nIn an extensive interview with L., conducted in 2007 by\nMaria Manuel Clementino and Jorge Picado in\nBraga (Portugal) [",
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"], L. says:"
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"At Reed I was instructed that the first year of calculus should\nconcentrate on foundations, formulas there being taught in the second\nyear. Therefore […] I spent several preparatory weeks trying to\ndevise a calculus course based on Zermelo–Fraenkel (ZF) set theory.\nHowever, a sober assessment showed that there are far too many layers of definitions,\nconcealing differentiation and integration from the cumulative hierarchy, to be able to\nget through those layers in a year. The category structure of Cantor’s structureless sets\nseemed both simpler and closer. Thus, the elementary theory of the category of sets arose from a purely practical educational need."
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"F. W. Lawvere, A. Heller, R. Lavendhomme (in the back) and A. Carboni at CT99, Coimbra, Portugal (© M. M. Clementino and J. Picado)"
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"Many of L.’s mathematical achievements (notions, constructions and theorems)\nresult from efforts to\nimprove the teaching of calculus and of engineering mathematics, and\nled him to conclude\nthat a workable foundation for mathematics, even for a calculus course,\ncannot be formulated in terms of ",
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" (and their composition). L. says,\nin the Braga interview [",
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"Philosophically, it may be said that these developments\nsupported the thesis that even in set theory and elementary\nmathematics it was also true as has long been felt in advanced algebra and topology, namely that the substance of mathematics resides not in Substance, as it is made to seem\nwhen ",
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" is the irreducible\npredicate, but in Form, as is clear when the guiding notion is\nisomorphism-invariant structure, as defined, for example, by universal\nmapping properties. As in algebra and topology, here again the\nconcrete technical machinery for the precise expression and\nefficient\nhandling of these ideas is provided by the Eilenberg–Mac Lane theory of categories, functors and natural transformations."
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"After the year at Reed College, L. went to Zürich, where he was visiting in 1964–66 at\nBeno Eckmann’s Forschungsinstitut für Mathematik. Eckmann had\nsucceeded in attracting several category\ntheorists to participate. Notably, the concept of ",
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"\n(“triple”), and its relationships to\nalgebraic theories and homology were elaborated (as documented in\n[",
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"])."
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"From Zürich, it was possible to attend seminars at the nearby Oberwolfach in South Germany.\nHere, L. met Peter Gabriel and learned from him aspects of\nGrothendieck’s approach to geometry,\nas expounded in SGA4\n[",
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"]."
]
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"3 Grothendieck"
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"id": "S3.p1",
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"Grothendieck’s work had a fundamental influence on L.’s later work.\nThey first met each other at the ICM in Nice (1970),\nwhere they both were invited lecturers. L. had here publicly disagreed with Grothendieck in a separate\nlecture on his “Survival” movement."
]
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"In 1973, they were both visiting\nBuffalo. L. says in the Braga interview:"
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"I vividly remember his tutoring me on basic insights of algebraic geometry,\nsuch as “points have automorphisms.”\nIn 1981 I visited him in his stone hut, in the middle of a\nlavender field in the south of France, to ask his opinion of a\nproject […].\nMy last meeting was at the same place in 1989 (Aurelio Carboni drove me there from Milano):\nhe was clearly glad to see me but would not speak, the result of a religious vow; he wrote on\npaper that he was also forbidden to discuss mathematics, though quickly his mathematical soul\ntriumphed, leaving me with some precious mathematical notes."
]
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"Lecturing in Coimbra, Portugal, March 1997 (© M. Sobral)"
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"4 Categorical dynamics and synthetic differential geometry"
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"In most of 1967, L. was assistant professor at the University of Chicago.\nL. here began to apply Grothendieck’s topos theory in an advanced lecture series,\ncentering around the problem of simplified foundations of continuum mechanics,\ninspired by Truesdell’s and Noll’s axiomatizations. The series was\nattended by Mac Lane, Jean Bénabou, Eduardo Dubuc and others, including\nthe present author, who was at that time finishing a thesis, under L.’s supervision.\nThe particular contribution which came out of the seminar was not yet a full-fledged\ncategorical dynamics, but a kinematic basis for it: the idea of having the tangent\nbundle construction representable, ",
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"\n(utilizing the Cartesian closed structure of the postulated category of spaces).\nAn aspect of this “kinematic” line of thought was later developed by several people as a more\nfull-fledged “synthetic differential geometry.”"
]
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"id": "S4.p2",
"content": [
"The wisdom from algebraic geometry, which was at the basis of this development\nin categorical dynamics, could also be imported and applied in standard smooth\ndifferential geometry; L. uses an algebraic theory (in the sense of his 1963 thesis),\nnamely the theory whose ",
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"-ary operations are the smooth functions ",
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" – a theory, for which it is crucial not to ask for a presentation in terms of generators and relations."
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"content": [
"5 Elementary toposes, algebraic geometry and logic"
]
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"L. returned to the Forschungsinstitut in Zürich in 1968–69. At this time, he had become\nmore convinced that toposes were involved not only as a background for categorical\ndynamics, but also for notions from set theory and logic: boolean-valued models,\nand forcing (as in Cohen’s work (1963) on the continuum hypothesis). In the Braga interview, he says:"
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"That these apparently totally different toposes, involving infinitesimal motion and advanced logic,\ncould be part of the same simple axiomatic theory, was a promise in my 1967\nChicago course. It only became reality after my second stay at the Forschungsinstitut\nin Zürich, Switzerland 1968–69, during which I discovered the nature of the\npower set functor in toposes as a result of investigating the problem of expressing\nin elementary terms the operation of forming the associated sheaf, and after\n1969–1970 […] through my collaboration with Myles Tierney."
]
}
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},
{
"type": "Paragraph",
"id": "S5.p2",
"content": [
"This collaboration took place in Halifax (Canada): In 1969,\nL. had obtained the prestigious Killam professorship at Dalhousie University in Halifax,\nand was in that context allowed to invite a dozen collaborators (among them Tierney),\nlikewise supported by Killam. This meant that during 1969–1971 Dalhousie became a lively place;\nmathematically, in particular, the notion of elementary topos gradually crystallized here.\nSignificantly, L. had organized that a preprint version of\n(exposé I–IV) of SGA4 [",
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"1"
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},
"]\nwas handed out to the participants of his seminar (SGA4 is Artin,\nGrothendieck and Verdier’s\n“Théorie des Topos et Cohomologie Etale des Schémas,” not officially published until 1972)."
]
},
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"id": "S5.p3",
"content": [
"However, in 1971, the dream team at Dalhousie was dispersed; the university administration\nrefused to renew the contract with L., due to his political activities in protesting against\nthe Vietnam war and against the War Measures Act proclaimed by\nTrudeau, suspending civil\nliberties under the pretext of danger of terrorism. (But in 1995, Dalhousie hosted the celebration of\n50 years of category theory, with participation of L.)"
]
},
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"type": "Paragraph",
"id": "S5.p4",
"content": [
"A conference organized by L., on the eve of his stay in Halifax in 1971,\ncarries the significant title: “Toposes, Algebraic Geometry and\nLogic,” and the proceedings from this conference were published in 1972\n[",
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"6"
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},
"]."
]
},
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"id": "S5.p5",
"content": [
"After leaving Halifax in 1971, L. became visiting professor in Aarhus (Denmark) 1971–72, and\nin Perugia (Italy) 1972–73. These were years where the new insights in topos theory,\nbrought about in Halifax, were consolidated and disseminated more widely. Also, after finally\nsettling 1973 in Buffalo (US), L. maintained close contacts, in the form of shorter and longer\nstays, with his European friends and collaborators; this includes a year 1980–81 at IHÉS (Paris)."
]
},
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"id": "S5.p6",
"content": [
"The toposes that we studied in Halifax and later, were in particular\n“gros toposes” (like the topos of simplicial sets),\nin contrast to the “petit toposes” (like the topos of sheaves on a topological space).\nThis was a distinction made in SGA4, IV.4.10. This distinction was for L. one of\nthe inputs of the study of the ",
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" of toposes, i.e., toposes in their functorial inter-relationship.\nSuch studies were developed by many researchers, and documented in many mathematical\nmonographs, articles,\nand in conferences (with or without proceedings). L. was very active in participating\nin conferences,\noften as invited keynote speaker; he was less active in getting the wealth of his ideas\nand visions\ndown in written form. For instance, his seminal talks in Chicago in 1967 on categorical dynamics\nwere not available in written form until in 1978, in the proceedings\nof a protracted “Open House”\nsummer meeting in Aarhus, on “Topos Theoretic Methods in Geometry”\n[",
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"]."
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"In 1982, L. (together with his Buffalo colleague Steve Schanuel)\norganized a conference in Buffalo,\n“Categories in Continuum Physics,” with participation also of many key researchers in continuum physics,\nlike Truesdell and Noll. Three of the articles in the proceedings\n(published in [",
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"8"
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},
"])\ndeal with the problem of foundations of thermodynamics."
]
},
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"type": "Paragraph",
"id": "S5.p8",
"content": [
"L. was in 1977 in the Scientific Steering Committee of the important\nand large summer meeting in\nDurham, “Applications of Sheaves” [",
{
"type": "Cite",
"target": "bib-bib4",
"content": [
"4"
]
},
"], which marked a breakthrough in\nexploiting the relative simplicity of toposes in the conceptualization of mathematical\nand physical theories. L. gave a talk in Durham on “categories in the foundations of thermodynamics,” of which, however, I have not been able to find a written account.\nThere does, on the other hand, exist accounts of a talk (with a lively debate) given by L. at this conference, with the title “The Logic of Mathematics,” where L. stated his\nview on the philosophy and development of mathematics. I include it here, since an obituary of L. would be incomplete, if it did not reflect the uncompromising character of\nhis political/philosophical life and work:"
]
},
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"In this Durham debate, L. says in the beginning of the talk (according to my notes and memory):"
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"content": [
"Mathematics is the science of space forms and quantitative relationships.\nWhat is the purpose of mathematics? Its purpose is to clarify this relationship in order to\nact as a basis of unity of people in solving problems (not mathematical problems) in the\nstruggle for production, and in the conscientiousness of this struggle,\nwhich is scientific experimentation."
]
}
]
},
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"id": "S5.p10",
"content": [
"Already at this early stage of the talk came an interrupting question (possibly rhetoric)\nfrom a member of the audience: “What is the purpose of production?”\nL. thought for quite some time before answering:\n“To bring you here!”"
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"The purpose of the logic of mathematics; to clarify and simplify the learning, use and\ndevelopment of mathematics. […] In a dialectical way: there is also a\ncounterpurpose:\nto obscure, complicate and prevent the learning, use and development of mathematics.\nIn particular, to freeze the development by promoting instead: thinking about forcing\neverything into a preconceived framework […].\nBoth of these purposes are fighting with each other inside each of us. […] Often the counterpurpose wins over the purpose.\nThis is because the counterpurpose is in the interest of the ruling class.\nThis is a thing which has changed drastically over the last 100 years. The interest of the monopoly capitalist class is against the development\nof production."
]
}
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"6 Axiomatic cohesion"
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"This is not the place to give (nor would I be able to give) a complete survey of\nall the aspects of L.’s mathematical and philosophical work. Just some further\nkey-words: probability, categorical logic, indexed/fibered categories, metric spaces as enriched\ncategories, linguistics, extensive vs. intensive quantities,\ncategory of physical quantities, Grassmann,\naxiomatic cohesion."
]
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"The idea of axiomatic cohesion, as introduced by L. 2007 [",
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"], has in particular\nled to recent new developments."
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"The following is a quotation from this 2007 publication:"
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"An explicit science of cohesion is needed to account for the varied background models for\ndynamical mathematical theories. Such a science needs to be sufficiently expressive to\nexplain how these backgrounds are so different from other mathematical categories, and\nalso different from one another, and yet so united that they can be mutually transformed.\nAn everyday example of such mutual transformation is the weatherman’s application of\nthe finite element method (which can be viewed as analysis in a combinatorial topos) to\nequations of continuum thermomechanics (which can be viewed as analysis in a smooth\ntopos, where smooth functions and distributions live)."
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"The basis for this axiomatic science of cohesion is a string of four functors"
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"each one in the string left adjoint to the next one. An example of such a string\nis familiar in topology: ",
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"id": "S6.p6",
"content": [
"Probably, many fruitful plants will emerge in the future from these seeds.\nThe germination of the seeds would be enhanced if they were more accessible\nin some archive. Some activity in creating such archives is taking place, notably in ",
{
"type": "Link",
"target": "www.acsu.buffalo.edu/~wlawvere",
"content": [
"www.acsu.buffalo.edu/~wlawvere"
]
},
"."
]
},
{
"type": "Paragraph",
"id": "authorinfo",
"content": [
"\nAnders Kock is emeritus professor of mathematics in the Department of Mathematics of University of Aarhus, Denmark. He graduated from the University of Aarhus in 1963 and studied for the Ph.D. under Lawvere in Chicago and Zürich in 1963–67. He was postdoc in Halifax in 1969–70, and collaborated with Lawvere in Aarhus in 1971–72. In May 1973, May 1978, and June 1983 he organized two-week Open House workshops in Aarhus, with the participation of Lawvere, and participated in numerous category theory conferences and workshops from 1966 until 2018. He is the author of several books, such as ",
{
"type": "Emphasis",
"content": [
"Synthetic Differential Geometry"
]
},
" (Cambridge University Press, 1981, 2nd ed. 2006) and ",
{
"type": "Emphasis",
"content": [
"Synthetic Geometry of Manifolds"
]
},
" (Cambridge University Press, 2010).\n",
{
"type": "Link",
"target": "mailto:kock@math.au.dk",
"content": [
"kock@math.au.dk"
]
}
]
}
]
}